A Framework to improve Turbulence Models using Full-field Inversion and Machine Learning

Abstract

Accurate prediction of turbulent flows remains a barrier to the widespread use of computational fluid dynamics in analysis and design. Since practical wall-bounded turbulent flows involve a very wide range of length and time scales, it is intractable to resolve all relevant scales, due to limitations in computational power. The usual tools for predictions, in order of their accuracy, includes direct numerical simulation (DNS), large-eddy simulation (LES), and Reynolds-averaged Navier-Stokes (RANS) based models.

DNS and LES will continue to be prohibitively expensive for analysis of high Reynolds number wall-bounded flows for at least two more decades and for much longer for design applications. At the same time, the high-quality data generated by such simulations provides detailed information about turbulence physics in affordable problems. Experimental measurements have the potential to offer limited data in more practical regimes. However, data from simulations and experiments are mostly used for validation, but not directly in model improvement.

This thesis presents a generalized framework of data-augmented modeling, which we refer to as field-inversion and machine-learning (FIML). FIML is utilized to develop augmentations to RANS-based models using data from DNS, LES or experiments. This framework involves the solution of multiple inverse problems to infer spatial discrepancies in a baseline turbulence model by minimizing the misfit between data and predictions. Solving the inverse problem to infer the spatial discrepancy field allows the use of a wide variety and fidelity of data. Inferring the field discrepancy using this approach connects the data and the turbulence model in a manner consistent with the underlying assumptions in the baseline model. Several such discrepancy fields are used as inputs to a machine learning procedure, which in turn reconstructs corrective functional forms in terms of local flow quantities. The machine-learned discrepancy is then embedded within existing turbulence closures, resulting in a partial differential equation/machine learning hybrid, and utilized for prediction.

The FIML framework is applied to augment the Spalart-Allmaras (SA) and the Wilcox's KOM model and for flows involving curvature, adverse pressure gradients, and separation. The value of the framework is demonstrated by augmenting the SA model for massively separated flows over airfoil using lift data for just one airfoil. The augmented SA model is able to accurately predict the surface pressure, the point of separation and the maximum lift -- even for Reynolds numbers and airfoil shapes not used for training the model. The portability of the augmented model is demonstrated by utilizing in-house finite-volume flow solver with FIML to develop augmentations and embedding them in a commercial finite-element solver. The implication is that the ML-augmented model can thus be used in a fashion that is similar to present-day turbulence model.

While the results presented in this thesis are limited to turbulence modeling, the FIML framework represents a general physics-constrained data-driven paradigm that can be applied to augment models governed by partial differential equations.